A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 3$ $a_{n+1} = 2a_n - c \quad (n > 1)$ where $c$ is a constant. (a) Write down an expression, in terms of $c$, for $a_2$ (b) Show that $a_3 = 12 - 3c$ (c) Given that $\sum_{i=1}^4 a_i \geq 23$ find the range of values of $c$.

A-Level Edexcel Maths Pure Differentiation: A sequence of numbers $a_1, a

A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 3$ $a_{n+1} = 2a_n - c \quad (n > 1)$ where $c$ is a constant - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 2

Question 7

$A-sequence-of-numbers-$a_1,-a_2,-a_3,-\ldots$-is-defined-by--$a_1-=-3$---$a_{n+1}-=-2a_n---c-\quad-(n->-1)$---where-$c$-is-a-constant-Edexcel-A-Level Maths Pure-Question 7-2012-Paper 2.png$

A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 3$ $a_{n+1} = 2a_n - c \quad (n > 1)$ where $c$ is a constant. (a) Write down an expression,... show full transcript

Worked Solution & Example Answer:A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 3$ $a_{n+1} = 2a_n - c \quad (n > 1)$ where $c$ is a constant - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 2

Step 1

Write down an expression, in terms of $c$, for $a_2$

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Answer

To find $a_2$ , we use the recurrence relation:

$a_2 = 2a_1 - c$ Substituting $a_1 = 3$ gives: $a_2 = 2(3) - c = 6 - c.$

Step 2

Show that $a_3 = 12 - 3c$

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Answer

Using the recurrence relation again to find $a_3$ :

$a_3 = 2a_2 - c.$ Substituting $a_2 = 6 - c$ :

$a_3 = 2(6 - c) - c = 12 - 2c - c = 12 - 3c.$

Step 3

Given that $\sum_{i=1}^4 a_i \geq 23$, find the range of values of $c$.

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101 rated

Answer

First, we compute $a_4$ using the recurrence relation:

$a_4 = 2a_3 - c = 2(12 - 3c) - c = 24 - 6c - c = 24 - 7c.$ Then, we find the sum:

$\sum_{i=1}^4 a_i = a_1 + a_2 + a_3 + a_4 = 3 + (6 - c) + (12 - 3c) + (24 - 7c)$
$= 45 - 11c.$ Setting this greater than or equal to 23:

$45 - 11c \geq 23$
Solving for $c$ gives:

$45 - 23 \geq 11c \Rightarrow 22 \geq 11c \Rightarrow c \leq 2.$
Thus, the range of values of $c$ that satisfies the condition is:

$c \leq 2.$

A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 3$ $a_{n+1} = 2a_n - c \quad (n > 1)$ where $c$ is a constant - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 2

Worked Solution & Example Answer:A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 3$ $a_{n+1} = 2a_n - c \quad (n > 1)$ where $c$ is a constant - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 2

Write down an expression, in terms of $c$, for $a_2$

Show that $a_3 = 12 - 3c$

Given that $\sum_{i=1}^4 a_i \geq 23$, find the range of values of $c$.

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